A geometric framework for the subfield problem of generic polynomials via Tschirnhausen transformation

Let $k$ be an arbitrary field. We study a general method to solve the subfield problem of generic polynomials for the symmetric groups over $k$ via Tschirnhausen transformation. Based on the general result in the former part, we give an explicit solution to the field isomorphism problem and the subfield problem of cubic generic polynomials for $\frak{S}_3$ and $C_3$ over $k$. As an application of the cubic case, we also give several sextic generic polynomials over $k$.